a search for deviations from newtonian gravity a quest in ... · we can visualize the three...

A search for deviations from Newtonian gravityA quest in a higher dimensional world

Shanna Haaker, Wouter Merbis and Loran de Vries

July 9, 2006

Abstract

Motivated by theoretical considerations involving the existence ofextra compact dimensions of space, mostly developed to solve thehierarchy problem, we explore the possibility to extend our 4D space-time with more spatial dimensions. After an introduction into the math-ematics of extra dimensional physics, we examine the corrections toNewton’s Gravitational law, inspired by this extra dimensional physics.These deviations can be described by Yukawa-type corrections. Wediscuss constraints placed on these corrections from recent short-range gravity experiments and theoretical constraints that arise fromastrophysics.

1

Contents

1 Introduction 2

2 Newton’s law of Gravitation in more than 4D 42.1 The ”volume” of a (d− 1)-dimensional sphere . . . . . . . . . 62.2 An application: Planetary orbits in higher dimensions . . . . . 9

3 Compactification and Kaluza-Klein Reduction 123.1 Compactification . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 KK-modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Kaluza-Klein reduction . . . . . . . . . . . . . . . . . . . . . . 14

4 Gravitational potential in n compactified dimensions 164.1 The exact gravitational potential for n = 1 . . . . . . . . . . . 18

5 Deviations from Newton’s inverse square law 205.1 Deviations from compactification on a n torus . . . . . . . . . 215.2 Deviations from the 1/r potential in a n-dimensional compact

manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Fundamental energy scales and the Hierarchy Problem 276.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2 Compactified dimensions to the rescue . . . . . . . . . . . . 29

7 Short-Range Tests of the gravitational inverse square law 327.1 Gravity measurement . . . . . . . . . . . . . . . . . . . . . . 327.2 The Eot-Wash experiment . . . . . . . . . . . . . . . . . . . . 337.3 The Colorado experiment . . . . . . . . . . . . . . . . . . . . 357.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.4.1 Eot-Wash group . . . . . . . . . . . . . . . . . . . . . 377.4.2 Colorado group . . . . . . . . . . . . . . . . . . . . . . 37

8 Astrophysical bounds 398.1 Cross section of graviton . . . . . . . . . . . . . . . . . . . . . 398.2 The Sun and red giants . . . . . . . . . . . . . . . . . . . . . 428.3 SN1987A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

9 Conclusion 46

CONTENTS 1

Preface

Over the last century the hectic world of theoretical physics has flourishedlike never before. The discovery of special relativity and the developmentof quantum mechanics paved the way for a chain of new theoretical break-troughs and discoveries. In a relatively short period of time, theories grewto give very precise descriptions of not only the world around us, but alsothe universe at scales that are unimaginable small or astronomically big.

One of the most recent discoveries in this ever expanding web of theoreti-cal physics is string theory. Without getting into too much detail about thispromising young theory, we will explore one of its features in this article:extra dimensions. To be more precise, we will explore some of the possi-bilities extra dimensions have to offer, but most of all: the constraints thatrestrict their existence.

This article was created as a part of the second year project for physicsstudents at the University of Amsterdam (UvA). Our project group originallyconsisted out of six second year students. Together we worked through acouple of exercise sets that made us calculate and fully grasp every stepmentioned in sections 2 - 5 and parts of section 6. These first couple ofsections will be more or less the same in their and our report, as we werefirst studying as a group of six. We would like to thank them for their help onthese sections. After the introduction we were divided into two groups. Onegroup went on to study another fascinating phenomena concerning extradimensions; the possibility to create miniature black holes in particle ac-celerators. We devoted this spring to the study of short-range gravity andindications that this subject can give us for the existence of extra dimen-sions.

We want to express our deep gratitude to Nick Jones, our project leader,for his patience, devotion and his ever clear explanations.

1 INTRODUCTION 2

1 Introduction

Before we begin our journey through the physics of higher dimensions, weshould make some things clear. To begin with: what do we mean with adimension?

You can define the number of dimensions as the number of independentdirections along which you can travel. When we look around us, we seeone direction in front of us, one aside of us and one above. We count threedirections and the conclusion is rapidly drawn that we must live in three di-mensions. You can also define the number of dimensions as the (precise)number of coordinates you need to completely pin down a point in space.Mathematicians use in this context the notation (x, y, z) and add a newsymbol for every new dimension. We can visualize the three dimensionswe just counted by three coordinate axes. One extra dimension simply re-quires adding another axis, independent of the first three axis. This created4D coordinate system may be hard to draw, but must not be hard to imag-ine. When the extra dimensions come along we better stop visualizing andcontinue with words and equations.

Everything around us, since the day we were born, suggests that we areliving in three spatial dimensions. This state of mind remains for almostevery person on earth. A small group, who is introduced to special rel-ativity, discovers the combination of our three dimensions of space withone dimension of time, forming the space-time continuum. In this constructspace and time become inseparable. This group of privileged people is of-fered time as the 4th dimension. Mathematically speaking this means thatyou need four coordinates to describe an event in space-time. Now we aregetting to the even smaller crowd of people who are considering even morespatial dimensions. This last gang consists of well educated people, mostof them physicists. These are normally not the kind of folks who would be-lieve in science fiction. What has gotten into them? Why seriously thinkabout extra dimensions that you cannot even see?

First we have to note that not having seen another dimension does notmean that there is none. In addition, no physical theory states that theremust be only three dimensions of space. Still, this does not explain whywe would go through a whole lot of trouble working out the mathematicsin a more dimensional world. The best motivation comes from the searchfor the holy grail of physics: unification. Our heavyweight theory of today

1 INTRODUCTION 3

is the standard model. This theory describes three out of four fundamentalforces: the strong, weak and electromagnetic forces. However the standardmodel is not a complete theory. The theory does not include the fourth fun-damental force: gravity.

To give an accurate description of gravity and how it behaves, we have toturn to Einstein’s general relativity. The problem is that this theory breaksdown at very short distances due to quantum mechanical effects. In addi-tion, relativity is uncapable to describe all of the other forces. The unificationof all the fundamental forces into one theory is the holy grail of physics: thefusion of Einstein’s general relativity, with the standard model. The only se-rious candidate that can make this dream come true is string theory. In thismodel of theoretical physics, particles and forces are presented as tiny ex-tended object: strings. It appears that string theory can only be consistentif there are up to seven additional spatial dimensions. [1]

But surely these extra dimensions cannot be of any shape or size? Whatare the constraints that we have to put on extra dimensions so that theirexistence does not contradict experimental evidence? The main focus ofthis article will be on the bounds that have to be put on extra dimensions,in order for them to stay consistent with astrological observations, experi-ments and theoretical frameworks.

In order to explain how these bounds can be obtained by experiment, wefirst have to explore the mathematical environment of higher dimensionalphysics to get an idea of what to measure. First we will discover how New-tonian gravity behaves in arbitrary number of dimensions. Then, in sections3 and 4 we will introduce compactification, the idea that the additional spa-tial dimensions are wrapped into a very small volume, and we will work outthe Newtonian gravitational potential in this new compactified space-timecontinuum. After this we turn to the hierarchy problem, which is believed tobe solved if we assume extra dimensions. This will give a good impressionon the consequences that extra dimensions have to our four-dimensionalworld. Finally, we turn to the experimental part of this article. We will con-sider various experiments, both laboratory work and astrophysical observa-tions. They all have one thing in common: they restrict the size of the extradimensions. At the end we will make a definite conclusion that will clearlyshow the conditions under which extra dimensions can exist.

2 NEWTON’S LAW OF GRAVITATION IN MORE THAN 4D 4

2 Newton’s law of Gravitation in more than 4D

From the preceding section we came up with the why of our extra dimen-sions quest. The next step we are going to take is looking at physics in aworld with more spatial dimensions than the three we are used to in every-day life.

Let’s consider Newton’s gravitational force law. This icon of classical physicstells us how the gravitational force depends on the distance, r, between twomassive objects m and M :

F (r) =GNmM

r2(1)

with GN the gravitational proportionality constant.

Every grown-up multidimensional theory which includes gravity should re-produce this formula. This will form a check point along the way. The man-ner in which this inverse square law depends on distance is strongly linkedwith the number of spatial dimensions. This number tells us how gravitydiffuses as it spreads in space. Before we adjust our formula to more thanthree spatial dimensions it might be nice to have some picture of how thisspreading takes place.

As a descriptive explanation we picture the problem of watering a plant ina garden. We distinguish between giving the plant the water through anozzle or through a sprinkler. In figure 1 we can see the differences be-tween the two methods. When we use the spout, all the water will end up

Figure 1: The amount of water delivered by the sprinkler is less than theamount delivered by the nozzle. (Figure from page 44 of [1])


Figure 2: The same number of field lines intersect a sphere of any radius(Figure from page 45 of [1])

on the plant, whereas using the sprinkler the water is spread. Accordinglynot every drop will reach the plant. Besides that, the distance between thesprinkler and the plant does matter as well. With the nozzle this is not thecase. The importance of this illustration is to recognize the fundamentaldistinction between the two watering methods, that is, jumping to a higherdimension. The spout only gives water to a point (one dimension), otherthan the sprinkler, which distributes the water on to a surface (two dimen-sions). In general we can say that anything that is spread in more than onedirection will have a lower impact on objects that are farther away. Similarly,gravity will spread more quickly with increasing distance.

We will represent the strength of gravity by field lines (in analogy with thesprinkler: the water flux). The line-density indicates the strength of the grav-itational force at a given point. Because of the fact that gravity attracts allthe surrounding mass isotropically, the field lines will go radially outwards.Therefore, as you can see from figure 2, the same number of field lines willintersect a sphere of any radius and the field lines become more diffusealong the way. Due to the fact that the fixed number of gravitational fieldlines are spread over a sphere’s surface, we can conclude that the gravita-tional flux has to decrease with the radius squared (see page 42 - 46 of [1]).

The mathematical description of the above is given by Gauss’ law, which,in a gravitational field, gives the relation between the gravitational flux flow-ing out of a closed surface and the mass enclosed by this surface. Wewill use Gauss’ law to derive Newton’s law of gravitation in more than fourdimensions (we are considering space-time):∫

surface

~g · d~S = −4πGNM (2)


where, ~g, is the acceleration due to gravity, caused by a point mass, M . Theintegral is taken over any surface which completely surrounds the mass.

To use this law we have to realize a few things. First we choose the surfacearound the point mass to be a sphere. Secondly, d~S is a unit vector pointingradially out of this sphere, whereas the direction of ~g is the negative radialdirection and thus they are antiparallel. Now the integral becomes easy tosolve, because we can pull out g (it has the same value everywhere on thesurface and additionally we can work with the norm) and the integral is justthe surface area of a sphere. Hence,

−g

∫surface

dS = −g · 4πr2 = −4πGNM (3)

To get equation (1) all there is left to do is substitute Newton’s second lawF = m · a with g as a and we obtain the desired result.

We are interested in expanding this relation to more dimensions. Assumingthat (2) holds for more than three dimensions, we can easily rewrite (1) ford dimensions in terms of the volume of a (d− 1)-dimensional sphere (note:(d− 1)-sphere’s live in d dimensions).Further, if (2) holds in d dimensions, than (3) must hold as well. For 3dimensions the integral on the left was just the surface area of a sphere,however now that we are in d dimensions it becomes the surface area of a(d−1)-sphere: Vd−1(r) (We shall refer to this as their “volume”). We obtain,

gVd−1(r) = 4πGNM ⇒ F =4πGNmM

Vd−1(r)(4)

The last hurdle we have to take in order to produce Newtonian gravity in ddimensions is evaluating the volume of a (d− 1)-dimensional sphere.

2.1 The ”volume” of a (d− 1)-dimensional sphere

Several times we were confronted with the surface or ”volume” of a hyper-sphere. Let’s look at this concept more precisely. A point in a d-dimensionalEuclidean space is represented by (x1, x2, . . . xd). The surface of a d-dimensional sphere of radius R is then defined by the equation:

x21 + x2

2 + . . . + x2d+1 = R2. (5)


For a 0-dimensional sphere (5) reads x21 = R2, so x1 = ±R. The volume

of a 0-dimensional sphere consists of two points at +R and −R, living in a1-dimensional world (we will not evaluate the “volume” just yet, but we shallsee later on that it is 2).

A 1-dimensional sphere is given by the equation x21 + x2

2 = R2. This is acircle in two dimensions and its volume can be obtained by integrating ainfinitesimal bit of the circle’s circumference Rdφ over the entire circle:

V1 =∫ 2π

0Rdφ = 2πR (6)

The 2-dimensional sphere is what we commonly hold for a sphere and itssurface is given by x2

1+x22+x2

3 = R2. The volume of this sphere can also beobtained by integration. Only one more variable is needed, because thereis one more dimension in this problem. We must integrate the infinitesimalbit R sin(θ)dθ over half a circle (the other integral will take it around thewhole sphere). We obtain:

V2 =∫ 2π

0Rdφ

∫ π

0R sin(θ)dθ = 4πR2 (7)

From all this we can make the generalization that the volume of a (d − 1)-sphere must depend on r(d−1). Moreover from (4) we see that the gravita-tional force F must depend on r−(d−1).We can ask ourselves how the preceding is applied to the volumes ofspheres with dimensions higher than 2? We will answer this by derivinga formula for Vd−1(r) in order to complete (4). To do this we use a trick. Wewill evaluate the integral I given by:

I =∫

all space

ddr e−r2

in two ways; in Cartesian coordinates and in polar coordinates.In Cartesian coordinates we know that r2 = x2

1 + x22 + . . . + x2

d. Since theintegral must be over all space, which is from −∞ to +∞ for every variablexi with 1 ≤ i ≤ d, the integral I becomes:

I =∫ +∞

−∞dx1dx2 . . . dxd e−(x2

1+x22+...+x2

d) (8)


This is the known integral∫∞−∞ dxe−x2

=√

π in d dimensions. Hence,

I =√

πd = π

d2 .

Now we will look at the problem in polar coordinates. To evaluate the inte-gral over all space is the same as taking the volume of a (d− 1)-sphere asa function of r and integrating that over all r′s (from 0 to ∞).

I =∫ ∞

0dr e−r2

Vd−1(r) (9)

We already know how Vd−1(r) depends on r and d (quantitatively: Vd−1(r) =c · rd−1, where c is a constant). We can substitute this into the equationabove.

I =∫ ∞

0dr e−r2

crd−1 (10)

We can solve this in terms of the Gamma function:

Γ(x) ≡∫ ∞

0tx−1e−tdt.

if we substitute t = r2, we obtain

I =c

2

∫ ∞

0dt e−tt

d2−1 (11)

(here we use that if t = r2, then dr = dt2r and r = t

12 ). The integral is just

the Gamma function for x = d2 and we already know from the Cartesian

coordinates what the answer should be.

I =c

2Γ(d/2) = π

d2 (12)

This can be solved for c and that gives us a result for Vd−1:

Vd−1(r) =2π

d2

Γ(d2)

rd−1. (13)

We can check this equation for d = 1, 2, 3 (remember: Γ(n) = (n−1)!, whenn is an integer, Γ(1/2) =

√π and Γ(3/2) =

√π/2). We find the same re-

sults we ran into earlier and now we see why the volume of a 0-dimensionalsphere equals 2. In the rest of the article we will be running into this ex-pression occasionally, but then we will refer to the volume of a unit sphere,


which we will also call Vd−1. So from here we will drop the r-dependence.

We have almost achieved our goal. We can substitute our expressionfor Vd−1(r) in equation (4) to end up with the expression for F in a d-dimensional space.

F =2πGNmMΓ(d

2)

πd2 rd−1

(14)

2.2 An application: Planetary orbits in higher dimensions

With the expression for the gravitational force in more dimensions, whichwe obtained in the preceding section, we can check if there can be sta-ble planetary orbits in higher dimensions. To do so, we consider the totalenergy of the planet:

E =12mv2 + V (r), (15)

where the gravitational potential V (r) is a function of r and alters whenwe change the number of spatial dimensions. By looking at the shapeof the potential, we can draw conclusions about the stability of planetaryorbits in higher dimensions. We can evaluate the potential by integratingthe gravitational force:

V (r) = −∫ ∞

rF (r)dr =

{k 1

(2−d)rd−2 d 6= 2k ln r d = 2

(16)

where k is a constant which is different for different dimensions:

k = 2πGNmMΓ( d2)

πd2

≥ 0

Because we are considering planetary orbits, we can be satisfied with a2-dimensional description of the orbit. We can therefore write ~r and ~v interms of 2-dimensional polar coordinates (r, θ). ~r becomes rr, while ~v (thetime derivative of ~r) becomes rr + rθθ.The angular momentum of the system must be constant, because we arelooking at an orbit.

~l = ~r × ~p = rr ×m(rr + rθθ) (17)


l = 0 + mr2θ ⇒ θ =l

mr2. (18)

If we substitute the potential and ~v = (rr + rθθ) in (15) we get the expres-sion:

E =12m(rr + rθθ)2 +

{k 1

(2−d)rd−2 d 6= 2k ln r d = 2

(19)

working out the quadratic factor and substituting(18) for θ we obtain:

E =12mr2 +

l2

2mr2+

{k 1

(2−d)rd−2 d 6= 2k ln r d = 2

(20)

We have reduced the problem to a one-dimensional situation (there is onlyr-dependance) and we have split (20) in two parts; one dependent of r,associated with the kinetic energy of the system and one dependent on r,associated with the potential of the system. In order to see if a planetaryorbit could be stable, we can look at the potential of the system in differentdimensions.

V (r) =l2

2mr2+

{k 1

(2−d)rd−2 d 6= 2k ln r d = 2

(21)

We plotted the curve for d = 3 and d = 5, they are shown in figure 3. Thispotential can only have a minimum value for d − 1 < 3. For higher dimen-sions than three there is either no extreme value (for d = 4) or a maximum,which means that planetary orbits are not stable in higher dimensions. They

Figure 3: A plot of the gravitational potential for d = 5 and d = 3 at the left,respectively the right.


could exist, but the slightest knock would put them out of orbit. The radialposition of the planets can not be stable.

This result should not be to surprising, since we have never seen a devia-tion to the inverse square law on ordinary distances. We know that New-ton’s law behaves as 1

r2 so this automatically leads to the conclusion thatd = 3. Also the assumption that the extra dimensions are of infinite sizesounds a little strange. If they were, why would we be uncapable to seethem, or take a walk in them?

3 COMPACTIFICATION AND KALUZA-KLEIN REDUCTION 12

3 Compactification and Kaluza-Klein Reduction

3.1 Compactification

In the preceding sections we came up with some mathematical descriptionof physics involving more spatial dimensions. Once more we can ask somequestions that already arose in the introduction. How is it possible that theuniverse could appear to have only three dimensions of space if the fun-damental underlying spacetime contains more? Why are they not visible?And if they are real, do these extra dimensions have any discernible impacton the world we see?

The idea of extra dimensions was first introduced in 1919 by the Polishmathematician Theodor Kaluza. He recognized the possibility of extra di-mensions in Einstein’s theory of relativity. The questions that are botheringus were the same questions asked then by Einstein. These questions re-mained unanswered until 1926, when the Swedish mathematician OskarKlein shed some light on the case. He provided a reasonable explanationto these problems, by imagining these extra dimensions to be compacti-fied, meaning that they are curled up into circles so tiny that we would notever suspect their existence (just 10−35m). This concept is now known asKaluza-Klein reduction. Thus in the Kaluza-Klein universe space can havenot only extended dimensions like the ones we are familiar with, but alsoextremely small curled-up dimensions.

Figure 4: The tight-rope artist can move in one direction, but the spidercan move in two directions along the cord. So the tight-rope appears one-dimensional for the artist and two-dimensional for the spider


In this section we will try to get a good idea on how these extra curled-updimensions look like and how the physics works out when we assume oneextra dimension to be compactified on a circle. We will see that in this newspace-time, gravity and the Coulomb force can be unified in one expression.

To begin with, how can we imagine space to be curled up? The best way tomake this clear might be by considering the one-dimensional world of thetight-rope artist. In figure 4 you can see that the tight-rope artist is restrictedto the tight-rope. He can move in only one direction, so the rope appears1-dimensional to him. However, if we zoom in on the tight-rope, we can seea little spider crawling along the cord. This spider is confined to the samerope, but he has two independent directions along which he can travel: hecan move along the length of the cord, or go around it in circles. The sametight-rope appears two-dimensional to the spider.

This example shows that a universe that has just one extended dimensionto large objects, might look multi-dimensional to small objects. We cangeneralize this picture to a universe with more extended dimensions, likeour well know four-dimensional world, which has three extended spatial di-mensions. This universe might also look more dimensional for very smallobjects, because the extra dimensions are curled up. Of course, thesecurled-up dimensions do not have to be in the shape of a circle, they canbe any shape you would like to imagine them, as long a they are periodic inspace. For a more detailed look into imagining compact extra dimensionssee chapter 8 of [2].

3.2 KK-modes

Let’s look a bit closer at the Kaluza-Klein compactification, where we as-sume one extra dimension to be curled-up on a circle of radius R. When weconsider one relativistic particle with mass m in this 5-dimensional world,the 5-momentum for this particle is given by:

p(5) ≡ (E

c, p1, p2, p3, p4). (22)

We know from special relativity that:

p(5) · p(5) = −E2

c2+ p2

1 + p22 + p2

3 + p24 = −m2c2. (23)


Where (p1, p2, p3) = p is the momentum in the three extended dimensionsand p4 is the momentum in the compactified dimension. We can rewritethis as an equation for the energy of the particle:

E2 = m2c4 + p2c2 + p24c

2. (24)

When we compare this equation to the equation for the energy in 4D:

E2

c2= M2c2 + p2 (25)

where M is the mass in four dimensions, we see there is an extra term inequation (23). This means that, since E and c are the same for both 4D en5D, the mass m in 5D has to be smaller than the mass M in 4D to compen-sate for the extra factor p2

4.

We can find an explicit expression for p4, using quantum mechanics. Thedirection of the particle in the curled-up dimension is periodic over time(x4 = x4 + 2πR). We therefore consider a particle stuck on a circle. Real-izing that the particle’s wavefuntion Ψ(x, p) ∝ e

ipx~ it must be single-valued.

Its wavelength has to fit a whole number of times on this circle. Hence,

eip4x4

~ = eip4(x4+2πR)

~ (26)

leads to:

p4 =n~R

(27)

We see that the momentum in the extra dimensions is proportional to 1R .

The situation were n = 1 or −1 is called the first Kaluza Klein modes (orshort: KK-mode). From the comparison of (24) and (25) we know that theKK-modes can be associated with mass. The momentum of the particle inthe curled-up dimension translates to mass in four dimensions.

3.3 Kaluza-Klein reduction

Now we observe two particles in 5D instead of just one. If we take the5-momenta of the two particles to be p and p′, then the gravitational forcebetween the particles, when they are at rest in the extended dimensions


(so p equals zero) and separated by r � R, can be approximated by thesemi-relativistic expression:

F (r) ' GN

c2

p · p′

r2. (28)

We can rewrite this force by substituting E = Mc2 (with M , the mass in theextended dimensions), p = 0 and equation (27) in:

p · p′ = −EE′

c2+ p · p′ + p4p

′4 (29)

so that (28) becomes:

F (r) = −GNM ′M

r2+

GNnn′~2

c2R2r2. (30)

In this equation we can identify both Newton’s gravitational force (the firstterm) and the Coulomb force (the second term). Notice that the secondterm can be both negative and positive (since n can be both as well), whichstates that this term can be both attractive and repulsive. To get all the con-stants right so that the second term is exactly the Coulomb force (which isgiven by Fc = qq′

4πε0r2 ), we would have to take n = n′ = 1 for the elementarycharge q and then work out what R must be. It turns out that the size ofthe extra dimension must be 1.8 · 10−34 m for the second term in (30) torepresent the Coulomb force!

4 GRAVITATIONAL POTENTIAL IN N COMPACTIFIED DIMENSIONS16

4 Gravitational potential in n compactified dimen-sions

The result we obtained in the last section that combined Newton’s gravityand Coulombs electromagnetic force in one expression is promising for fur-ther unification. This provides an excellent motivation for considering thegravitational potential in more than one compactified dimension.

In section 2.1 en 2.2 we have derived Newton’s gravitational potential in dspatial dimensions (equation 16):

V (r) = k1

rd−2(31)

where k = 2πGNmMΓ( d2)

(2−d)πd2

. We can express the potential in 4 + n space-

time dimensions as, using G4+n: Newton’s gravitational constant in n + 4spacetime dimensions:

V (r) = −G4+nmM

rn+1(32)

We now turn to the question what the gravitational potential would look like,if the additional n dimensions were compactified on to a circle with radius R.We can imagine this as we consider a mass M on a cylinder. The lengthof the cylinder represents the three unfolded spatial dimensions and theradius the n compactified dimensions. We can solve the problem using themethod of images (just like the electrodynamic problem of a point chargeon a cylinder).If we first imagine n to be 1, we can unfold the extra dimension, so to getan infinite extra dimension, with the mass M repeated every 2πR. Equation(32) now becomes an infinite sum over all the masses. The distance to themass becomes

√r2 + (b2πR)2, where b is an integer going from −∞ to∞.

Following this reasoning, the potential becomes:

V (r) = −∞∑

b=−∞

Gn+4mM

[r2 + (b2πR)2]12

(33)

If we generalize this to n compactified dimensions, we get an expression interms of n infinite sums:

V (r) = −∞∑

b1=−∞· · ·

∞∑bn=−∞

Gn+4mM

[r2 + (b12πR)2 + · · ·+ (bn2πR)2]n+1

2

(34)


In the limit of r � R these sums can be replaced by integrals. This isbecause the pieces of 2πR are small in comparison to r, that it can beapproximated as infinitely. Now (34) can be written as:

V (r) = −∞∫

b1=−∞

· · ·∞∫

bn=−∞

Gn+4mM

[r2 + (b12πR)2 + · · ·+ (bn2πR)2]n+1

2

db1 . . . dbn

(35)

To clean up this expression we divide the denominator by r2 and substitutexi = bi2πR

r to get:

V (r) = −Gn+4mM

r(2πR)n

∞∫x1=−∞

· · ·∞∫

xn=−∞

1

(1 + x21 + · · ·+ x2

n)n+1

2

dx1 . . . dxn

(36)

Now we can switch to polar coordinates using the volume of a n− 1 dimen-sional sphere (Vn−1(ρ)). The integral can be written (We will use ρ for theradial variable, because we already have a different r in the expression):

V (r) = −Gn+4mM

r(2πR)n

∫ ∞

0Vn−1(ρ)

1

(1 + ρ2)n+1

2

dρ (37)

If we now substitute u for ρ2, we can solve this in terms of the Beta function(from [17]);

B(p + 1, q + 1) =∫ ∞

0

updu

(1 + u)p+q+2=

Γ(p + 1)Γ(q + 1)Γ(p + q + 2)

(38)

Working out the integral and writing the Beta function in terms of Gamma-functions yields:

V (r) = −Vn−1Gn+4mM

2Σn

1r

(39)

where Vn is the volume of a n dimensional unit sphere given by (13) andΣn is the volume of the extra dimensions (in this case Σn = (2πR)n). Theequation looks just like the gravitational force we know in 4D. This is whatwe would expect, since we are working in the limit of r � R. We canacquire an expression for Gn+4 in terms of GN 39. We obtain,

Gn+4 =2GNΣn

Vn−1(40)


If we assume the extra n dimensions to be compactified on a circle, weobtain the usual inverse square law for distances much greater than thesize of the extra dimensions. However, if we look at distances close to R,the approximation of a sum by an integral no longer stands. Therefore wewould expect the gravitational force to deviate from Newtonian gravity andpick up an extra correction term. These are often referred to as Yukawa-type corrections (see, for instance [6]).

V (r) ∼ 1r(1 + αe

−rλ + . . .) (41)

4.1 The exact gravitational potential for n = 1

If we take n to be 1, we can work out (34), obtaining the exact potential.We can then check that the expression for the potential satisfies our expec-tations. If we calculate the limit of r � R we want to find the 1

r potential,while taking the other limit (R � r) we would expect the potential, derivedin section 2.2 to appear. In the last case we are looking at such small dis-tances that the extra dimensions appear very large. For n = 1 equation(34) becomes:

V (r) = −∞∑

b=−∞

G4+1mM

r2 + (b2πR)2(42)

Now we divide the denominator by 2πR and use the identity:

∞∑m=−∞

1m2 + a2

=π

acoth(πa). (43)

to get:

V (r) = −G4+1mM

2rRcoth

(r

2R

)(44)

We will first check the limit R � r, so we will need the limit limx↓0 coth x =2+x2

2x , where we can forget about the quadratic term, because in this case itwould be extremely small.

V (r) = −G4+1mM

r2(45)

which is exactly what we would expect!


Now we use the limit limx→∞ coth x = 1 to get the other limit:

V (r) = −G4+1mM

2Rr(46)

which is exactly (39) for n = 1.We have seen that for n = 1 the potential satisfies what we would physicallyexpect. The exponential power in (41) can be obtained by calculating thefirst order of corrections to equation (46). We can calculate this corrections,by keeping the first correction term of the approximation we made to getfrom (44) to (46). To simplify matters, we write

V (r) ∼ 1r

coth(

r

2R

)(47)

When we work out coth and multiply by e−r2R we obtain

V (r) ∼ 1r

(1 + e

−rR

1− e−rR

)(48)

In the limit where r � R, the term e−rR goes to zero. This is where we made

the approximation (see [18])

V (r) ∼ 1r(1 + e

−rR )(1 + e

−rR + . . .) (49)

Working out the brackets and omitting the last term leaves us with

V (r) ∼ 1r(1 + 2e

−rR + . . .) (50)

which is essentially the result stated in 41.

5 DEVIATIONS FROM NEWTON’S INVERSE SQUARE LAW 20

5 Deviations from Newton’s inverse square law

In the preceding section we have derived an expression for the gravitationalpotential in n compact dimensions. We have also stated that, looking at theproblem in four dimensions, this leads to a deviation from the 1

r Newtonianpotential for distances close to the size of the extra dimensions. This newpotential is given by:

V (r) ∼ 1r(1 + αe

−rλ ) (51)

This is what we stated in equation (41) (see, for instance [6]).

This equation is very useful for experimental physicists, because it givesa description of how the gravitational potential would change if there werecompactified extra dimensions. If someone would be able to measure thisdeviation from Newton’s law, this would be a strong proof that extra dimen-sions do in fact exist.

What is the physical interpretation of this deviation? Moreover, why is it anexponential power, and not some factor of 1

rn+1 as in equation (32)?

To answer the first question is that we just have to look closely at the formulaabove. The exponential power tells us that gravity will become stronger atshorter distances. Here we can see that the parameter α accounts for thestrength of the deviation. It tells you how much stronger it gets. While theλ parameter tells you at what distance the deviation begins to have effect.Intuitively, this parameter is related to the size of the extra dimensions, be-cause the deviation only starts to have effects at distances close to the sizeof the extra dimension.

We can try to measure these two parameters in order to put experimentalconstraints on the size of the compactified dimensions. High precision shortrange measurements of the inverse square law that do not show any devi-ation from the gravitational force or potential give us strong upper boundson α and λ and thereby on the size of the extra dimensions. We can alsoget experimental bounds on the size of extra dimensions from astrophysicalconstraints, as we will show in section 8.

The rest of this section is devoted to the answer of the second questionmentioned above. We will derive (51) in two ways. First we consider the


case of n extra dimension compactified on a circle (or torus) as we did insection 4. This will give us theoretical values for α and λ. Next we will con-sider the more general case, where the geometry of the extra dimensionsis unknown. Research on this subject is done in [4].

5.1 Deviations from compactification on a n torus

Just like before we consider 4 + n dimensional space-time and we let theextra n dimensions to be compactified on a circle of radius R. We assumeall the extra dimensions to be of the same size, although it does not changeanything in the derivation if they are not (then you just have to read Ri wherewe put R). As shown in section 4 the gravitational potential is given by ninfinite sums (see equation (34)).

Unlike before, now we consider not only the points in the same 4D directionas the mass M , but we also consider points that have a direction in theextra dimensions. We write {xi}n

i=1 = x for the direction vector in the extradimensions. It is not that hard to see that (34) now becomes:

V (r) = −∑b∈Z

Gn+4mM

[r2 +∑n

i=1(xi − 2πRbi)2]n+1

2

(52)

(see (2) of [4])where we can consider b = {b1, b2, . . . , bn} to be a vectorin the n-dimensional lattice, that is: every coefficient of b can take everyinteger number.

The trick we use to evaluate these sums is called Poisson resummation. Itallows you to rewrite the sum over a periodic function f(nR) with n ∈ Z toa sum over its Fourier transform f (from [15]).

∞∑n=−∞

f(nR) =∞∑

n=−∞

f( nR)

2πR(53)

We will apply Poisson resummation to our potential in (52) and althoughour function is periodic in n directions, the procedure is straightforward.Realizing that:

f(m) =∫

dnyGn+4mMe−im·y

[r2 +∑n

i=1(xi − yi)2]n+1

2

(54)


where m = { b1R , b2

R , · · · , bnR } and y is a vector in n space (y = {yi}n

i=1). Wesee from (53) that our potential becomes:

V (r) = −Gn+4mM

Σn

∑b∈Z

∫dny

e−im·y

[r2 +∑n

i=1(xi − yi)2]n+1

2

(55)

(see (4) of [4]) This can be evaluated if we shift the y coordinate to y + x(which is allowed because we integrate over all space) and then changeto polar coordinates. When we change to polar coordinates, we have torealize that the exponential depends on θ, where θ is the angle betweenm and y. This is because its argument hold an inner product betweentwo vectors in n space. Because the function holds a θ dependance, thechange to polar coordinates is given by:∫

dny =∫ ∞

0dρ

∫ π

−πdθρn−1(sin θ)n−2Vn−2 (56)

with Vn−2 the volume of a (n − 2)-dimensional unit sphere and ρ = |y| =√y21 + . . . + y2

n. It follows that:


Σn

∑b∈Z

e−im·x∫ ∞

0dρ

ρn−1

[r2 + ρ2]n+1

2

∫ π

−πdθe−i|m|ρ cos θ(sin θ)n−2

(57)

Substituting u (and du = sin θdθ) for cos θ and taking the real part gives:


Σn

∑b∈Z

e−im·x∫ ∞

0dρ

ρn−1

[r2 + ρ2]n+1

2

∫ 1

−1du cos(|m|ρu)(1− u2)

n−32

(58)

(see (7) of [4]) Note that |m| = ( b21R2 + b22

R2 + · · · + b2nR2 ) are the masses of

the KK-modes derived in section 3.2. This equation might seem a handful,however, by use of the following integrals ( from 8.411(8), 6.565(3) of [14])it is quite easy to get to the final answer.

Jν(z) =( z2)ν

Γ(ν + 12)Γ(1

2)

∫ 1

−1(1− t2)ν− 1

2 cos(zt)dt (59)∫ ∞

0xν+1(x2 + a2)−ν− 3

2 Jν(bx)dx =bν√π

2ν+1aeabΓ(ν + 32)

(60)


here Jν is the Bessel function of order ν. After performing the integrals wefind:

V (r) = −G4mM

r

∑b∈Z

e−r|m|e−im·x (61)

with G4 defined in (40).

At this point we need to remember that we are searching for an expressionfor how gravity would behave in 4D, assuming extra toroidal compactifieddimensions. Since all point particles in four-dimensional space-time can betaken to have x = 0, we get to the four dimensional gravitational potential:

V4(r) = −G4mM

r

∑b∈Z

e−r|m| (62)

(see (9) of [4]) Obviously, the term with b = 0 results in Newton’s expres-sion. As stated above, the second term (the one with |b| = 1) can beassociated with the lightest KK-mode. There are 2n of these modes (twoin every extra dimensions, since bi can be either 1 or −1.) and all of themhave mass 1

R . Thus the gravitational potential can be approximated by:

V4(r) = −G4mM

r(1 + 2n e−r/R) (63)

Here we see that for compactification on a n-torus, we get explicit values ofα and λ in equation (51). Especially nice is the fact that λ = R in this case,thus the deviations from the 1

r potential are starting to have a noticeableeffect at distances in the order of R. In the case of n = 2 we will showin section 6.2 that R must be in the order of millimeters. Therefore, sub-millimeter tests of the inverse square law must be able to strongly boundthe existence of two extra toroidal dimensions.

5.2 Deviations from the 1/r potential in a n-dimensional com-pact manifold

We have no idea how the extra dimensions would look like, if they exist atall. It is easy to picture them as being rolled up into a finite size, but for allwe know it might be a very complex geometrical shape. However, withoutsaying something about the precise geometry of the extra dimensions, wecan still work out an expression for the 4-dimensional potential that gives a


correction to Newton’s potential.

In the Newtonian limit the 4 + n potential obeys the Poisson equation (thisis of course in 3 + n dimensions, since the Laplacian does not cover time):

∇23+nV4+n = (n + 1)Vn+2G4+nMmδ(n+3)(x) (64)

(see (14) of [4])In a flat and uncompactified space, this equation is solvedby (32), but we are looking for an expression for V4+n with n dimensionson a compact manifold Mn. We do not know the precise geometry of then-dimensional compact manifold, but we can still define a set of functions{Ψm} as eigenfunctions of the Laplacian operator in n dimensions.

∇2nΨm = −µ2

mΨm (65)

(see (13) of [4])This set is orthogonal:∫Mn

Ψn(x)Ψ∗m(x) = δn,m, (66)

(see (11) of [4]) and complete. (n and m are vectors the n-dimensionallattice, just like b was in the last section.)

Now the trick is to solve equation (64) for V4+n with n dimensions compacti-fied by separation of variables. We will expand V4+n in terms of the basis ofeigenfunctions of the Laplace operator in the n-dimensional compact man-ifold, {Ψm}.

V4+n =∑m

Φm(r)Ψm(x) (67)

(see (15) of [4]) Note that Φm is only dependent of r. Substituting this inequation (64) yields for the left hand side:∑

m

[Ψm(x)∇23Φm(r) + Φm(r)∇2

nΨm(x)] (68)∑m

Ψm(x)[∇23Φm(r)− µ2

mΦm(r)] (69)

Now we use the orthogonality of {Ψm} by multiplying both sides of (64)with

∫Mn Ψ∗

m(x). We obtain:


∇23Φm(r)− µ2

mΦm(r) =∫Mn Ψ∗

m(x)(n + 1)Vn+2Gn+4Mmδ(n+3)(x)

= (n + 1)Vn+2Ψ∗m(0)Gn+4Mmδ(3)(r)

(for the above two lines of equations see (16) of [4])To solve this equation,you can fill in Φm as a Fourier transform: Φm(r) =

∫d3kΦm(k) eik·r. The

Fourier transform of the delta function equals one, so we get an expressionof Φm(k).

Φm(k) = −(n + 1)Vn+2Ψ∗m(0)Gn+4Mm

k2 + µ2m

(70)

Plugging this back into the definition of the Fourier transform would give usan expression for Φm(r), we would just have to solve:

Φm(r) = −(n + 1)Vn+2Ψ∗m(0)Gn+4Mm

∫d3k

eik·r

k2 + µ2m

(71)

Once again we can change to polar coordinates and pay attention to theinner product in the exponential, it will give us a cos θ term. We have lookedup the final integral over k and found for Φm(r) (from 3.723(3) of [14]):

Φm(r) = −VnΨ∗m(0)Gn+4Mm

21re−µmr (72)

(see (17) of [4]) Now equation (67) looks like:

Vn+4 = −VnGn+4Mm

2r

∑m

Ψ∗m(0)Ψm(x)e−µmr (73)

(see (18) of [4]) Just like before, we can set x = 0. Also like before, wewould expect some of the exponential powers in the sums to be the same.Remember that we had a degeneracy of 2n for the first KK-mode in thecase of the n-torus: we could go around the torus in one direction and theother way around. Consequently, symmetry in the compact manifold Mn

would also lead to some kind of degeneracy. We can change this sumabove to a sum which is only over all the irreducible representations mir

Vn = −GnMm

r

∑mir

dmire−µmir r (74)

(see (19) of [4]) We have used the theoretical result that the sum of |Ψmir |2over all representatives equals dmir/Σn. Now Σn 6= (2πR)n, but it gives the


volume of the compact manifold Mn. Additionally, dmir is the degeneracyof each irreducible representation.

This is about as far as it goes without getting into the actual shape of Mn.However, if you have a nice idea in mind for the shape of your extra dimen-sions, all you have to do is calculate the eigenvalues of the Laplace operatorµm and its degeneracy dm, and you will know the theoretical values for αand λ.

6 FUNDAMENTAL ENERGY SCALES AND THE HIERARCHY PROBLEM27

6 Fundamental energy scales and the Hierarchy Prob-lem

Before we dive into the experimental evidence for constraints on large extradimensions, we will take a look at some important energy scales at whichphysical processes take place. We will see that the fundamental scale atwhich gravity becomes strong is much bigger than the fundamental scaleat which the other forces operate. We will show in section 6.2 how extra di-mensions can nullify this difference, but first we take a look at these energyscales and how they are related to length.

6.1 Terminology

To get a good idea of the relation between energy and length, picture thefollowing. Only particles whose wavefunctions differ over very small scaleswill be affected by physical processes taking place at short distance. How-ever, when we recall the de Broglie relation, particles whose wavefunctioninvolve short wavelengths also have large momenta. Consequently we canconclude that you need high momenta, and hence high energies, to be sen-sitive to the physics of short distances ([1] page 143 ).

Therefore, particle physicists use the term energy scale for the place on theladder of energy that indicates the amount of energy they need to probe thephysical processes they want to study. We can convert an energy into a cor-responding length with the formula E = hc

λ and into mass using E = mc2.This length we just acquired illustrates the range of the associated force.The ”ladder of energy” just mentioned is pictured in figure 5 along with thecorresponding length scale. We see that climbing up the ladder in energy,means climbing down in length.

Now we have had some conceptual thought of energy scales, we shouldwrite down equations. The fundamental units GN , ~ and c can be combinedinto a new quantity with units of mass, and another with units of length.These quantities are called the Planck mass (Mp) and the Planck length(lp):

Mp ≡√

~c

GN= 2.2 · 10−8kg ∼ 1.2 · 1028eV. (75)


Figure 5: Some important length and energy scales in particle physics.(Figure from page 136 of [1])

and:

lp ≡√

~GN

c3= 1.6 · 10−35m (76)

We see from equation (75) that Newton’s gravitational constant is inverselyproportional to the Planck mass and thus also related in this way to a con-cept called Planck energy.

In nature there are at least two fundamental energy scales. On the onehand we have the electroweak energy scale (mEW ∼ 102 GeV ). This isthe scale at which the Standard Model operates, it determines the massof the elementary particles and above this scale the symmetry associatedwith the Standard Model is spontaneously broken. Current experiments inparticle accelerators are operating around this energy and have found re-sults verifying the Standard Model with great precision. When the LHC willbecome operational in 2007 at CERN we will be able to do experimentsabove the weak scale energy.


On the other hand we have the already mentioned Planck scale energy(Mp ∼ 1019 GeV ). We saw that Newton’s gravitational constant is in-versely proportional to Planck energy. Therefore gravity is weak becausethe Planck scale energy is large. Moreover the Planck scale energy is theamount of energy that particles would need to have for gravity to be a strongforce. As told before we can convert an energy scale into a correspondinglength scale, which tells us about the range of the force in question. Theenormous gap (note the 16 orders of magnitude difference in figure5) be-tween the Plank scale and the electroweak scale bothers a lot of physicists,and they have called this the hierarchy problem.

The supporters of the Grand Unified Theory like to think of gathering allphysics in one theory. However you can expect particles that experiencesimilar forces, to be somewhat similar. The enormous desert in betweenthe two energy scales does not help them much. For example, this gapgives a huge dissimilarity in the mass of the particles. Therefore manyphysicists take a lot of care in solving this mystery.

6.2 Compactified dimensions to the rescue

One possible solution of the hierarchy problem was proposed by Arkani-Hamed, Dimopoulos and Dvali in [3] (further we will refer to them as ADD).They propose a model where the weakness of gravity is explained by as-suming extra compactified spatial dimension. In their reasoning the gravi-tational and Standard Model interactions become united at the weak scale,which will then be the only fundamental scale in physics. The Planck scalewould only ”look” large to us, because it can spread into the compactifieddimensions.

Before we take a look at the Planck mass and length in 4 + n dimensionsand see how the Planck mass evolves, we must say something about theStandard Model forces in 4 + n dimensions. The Standard Model is a the-ory which works in four dimensions. In order not to mess with the elec-troweak scale, but to bring down only the Planck scale, we must assumethe Standard Model particles and forces to be confined to our 4-dimensionalworld. We can picture them as being stuck to a 4-dimensional membrane(or ”brane”) that lives in a (4 + n)-dimensional universe. Gravity is not con-fined to this brane and free to spread in the extra dimensions. This is whythe Planck scale seems much bigger than it is (remember from section 3.2


that momentum in the extra dimensions translates to additional mass in the4-dimensional world).

With the above in mind we must go back to the compactified dimensions.We can generalize the quantities from the beginning of this section to comeup with a Planck length and mass in n dimensions. We already know fromsection 4 that Newton’s constant has different units in more dimensions. In4 dimensions (that is: three spatial and one time dimension) it has units ofm3s−2kg−1. In n + 4 dimensions G4+n has units of mn+3s−2kg−1. You caneasily check that the Planck length now becomes:

lp =(

~G4+n

c3

) 1n+3

(77)

Now we can find an expression for the Planck mass in 4D (75), in terms ofthe Plank mass in (4 + n) dimensions (M4+n), given by:

M4+n =(

~n+1

G4+ncn−1

) 1n+2

(78)

Substituting Gn+4 from (40) in (78) and using (75) works out to be (we willdenote Mp as M4 for the 4 dimensional Planck mass to avoid confusion):

M24 =

(M4+n

)n+2 2Σncn

Vn−1(~n)(79)

With this result we can work out how large the extra dimensions must be asa function of the number of extra dimensions, if we know the value of M4+n.If we rewrite (79), pull out Rn from Σn, use the volume of a unit sphere andfill in M4 = 1016TeV, which we know from measurements, we get an explicitformula of R in terms of n and M2+n

4+n :

R =10

32n TeV

2n

M1+ 2

n4+n 2 n

√π

n−12 Γ(n+1

2 )(80)

If we write M1+ 2

n4+n in TeV , (80) has dimensions TeV −1. This comes from

the convention to set ~ = c = 1. To express R in meters we use E = 2π~cλ ,

this gives us:


R =1.98

2 n

√π

n−12 Γ(n+1

2 )

1032n−19

M1+ 2

n4+n

m (81)

Only n < 7 are the relevant extra dimensions for our purpose, since thisis the maximum known number of extra dimensions still consistent with atheory on physics in extra dimensions. The factor 1.98

2nq

πn−1

2 Γ(n+12

)can be left

out, since we only want to get an indication of the size of the possible extradimensions. Hence,

R ∼ 1032n−19

M1+ 2

n4+n

m (82)

Remember that M4+n has to be given in TeV . The first way to put this re-lation into practice is by assuming M4+n = mEW ≈ 1TeV, as proposed byADD. This is the value M4+n should have to solve the hierarchy problem.With this we can calculate the size of the extra dimensions:

n 1 2 3 4 5 6R(m) 1013 10−3 10−9 10−11 10−13 10−14

From the table above we can draw some conclusions. For n = 1 we havea size for the extra dimensions of R ∼ 1013 m implying deviations fromNewtonian gravity over solar system distances, so this case is ruled out.Gravity is very well studied over this kind of distances. If this deviationwould be true, then we would have noticed this. However, for all n ≥ 2the corrections to gravity only become noticeable at distances smaller thanthose currently probed by experiment. Recent experiments actually allowus to study gravity at smaller distance scales than ever before. In the nextsection we will examine two experiments of this kind. They will help us tosay something about n = 2.

7 SHORT-RANGE TESTS OF THE GRAVITATIONAL INVERSE SQUARE LAW32

7 Short-Range Tests of the gravitational inverse squarelaw

7.1 Gravity measurement

Gravity was the first of the four known fundamental forces to be understoodquantitatively since 1687. Despite this fact, gravity cannot compete in anyway to the detailed study of the other three forces at short distances. Thereis an evident lack of experimental data in this range. More concrete, untilrecently, nothing could be said about gravity for distances below 1 mm!

In the preceding sections we have brought the weakness of gravity severaltimes to your attention. The moment you want to carry out an experimentconcerning the strength of gravity at short distances, you walk right intoserious technical difficulties due to this weakness. To illustrate this oncemore: in any gravitational experiment the ’cancellation’ of the electromag-netic interaction between test masses must be at the level of roughly onepart in 1040 to leave any sensitivity to gravity. On short distance scales, lo-cal charge inhomogeneities and magnetic impurities in the materials of theexperiment quickly become important. A careful analysis of subtle system-atic effects is therefore crucial for any measurement of gravity at this scale.

Fortunately, experimental physicists like a challenge and came up with sev-eral ways to tackle this problem. This was not an easy task. The mainreason for giving this subject some serious thought came from the specu-lations about potential deviations from Newtonian gravity at short distancesas discussed in section 5.2. This adds yet another difficulty, because devia-tions from something weak must be even weaker. In the following sectionswe will describe some ingenious experiments that probe the gravitationalinteractions below the millimetre.

These experiments can be divided into two groups. The first group consistsof ”low frequency” experiments. In this category fall (among others) theclassic torsion balance used by Cavendish and the experiment describedbelow, carried out by the Eot-wash group. The name of this category ischosen only in contrast with the other one. In this second category fall the”high frequency” experiments. We will also describe one of them briefly.In this experiment, they use a kilohertz resonant-oscillator technique. Nowthe choice of the categories is clear. We used the articles of the variousresearch groups [5]-[8].


Figure 6: The apparatus of the Eot-Wash group. The distance between thependulum and the attractors has been exaggerated. (Figure from [9])

7.2 The Eot-Wash experiment

The first experiment we will encounter comes from the research carried outby the Eot-Wash group. In this experiment they used a torsion pendulum.A schematic representation is shown in figure 6.The setup consists of a ring suspended by a fiber above two disks (the at-tractors). During the experiment the two disks are set into slow rotationalmotion. As all torsion pendulums, the pendulum oscillates (i.e. twists anduntwists) after being given an initial torque When these attractors rotate,they cause the ring to twist back and forth due to gravitational interaction.The ring and the two disks have ten cylindrical holes drilled into them evenlyspaced about the azimuth. These holes of ”missing mass” are the key-elements of this experiment. When the holes would not be there, gravityfrom the disks would pull directly down on the ring and therefore would notbe able to twist it. The holes in the lower attractor are placed 18◦ rotatedcompared to those in the upper one. In this way they are rotated ”out ofphase”. This means that they lie halfway between the holes in the upperdisk as can be seen from figure 7. By attaching this second thicker disk inthis intelligent way, the lower holes produce a torque on the ring that sub-stantially reduces the signal due to Newtonian gravity.

However, now the holes are we may be able to measure an extra damping


Figure 7: Top view of the attractors. The solid circles represent the in-phase upper-plate holes. The open circles show the ones out-of-phase.Figure from page 6 of [5]

force, due to other additional forces. According to the theory described insection 5, there may be deviations from Newton’s gravitational force law ofthe form 63. Note that these correction terms make the gravitational forcestronger than predicted by Newtonian theory alone. In addition, they will dieaway very quickly with increasing distance, because they are exponentialfunctions.

As mentioned earlier the second disk significantly reduces the Newtoniangravity. However, when we assume that the deviations are correct, than,if gravity becomes stronger at short distances due to this deviations, thetorque induced by the lower disk to cancel the torque from the upper diskwill be relatively smaller than expected from Newtonian gravity alone. Thiswill happen because the lower attractor is farther from the pendulum ring.However, torques from a short-range interaction with a length-scale lessthan the thickness of the upper attractor disk will not be canceled. Thisresults in an extra damping force on the pendulum, due to the possibleYukawa interactions. Thus the geometry reduces the torque from Newto-nian gravity. but will have little effect on a induced short-range torque.

In order to diminish other forces, such as the electrostatic interactions be-tween the attractor and the pendulum, they put a stiff conducting membranebetween them. Additionally the pendulum was surrounded by an almostcomplete Faraday cage.

The amount of torque was measured by shining a laser beam on a mirrorinstalled on the pendulum, as can be seen from figure 6. The variation of


the magnitude of the torque with changing separation between the ring andthe disks provided a signature for any new gravitational or other short-rangephenomena. These measured torques were then compared to calculationsof the expected Newtonian and possible damping Yukawa effects. Fromthis they could say something about α and λ from equation (41). (For thissubsection ([9] was used)

7.3 The Colorado experiment

Figure 8: Active components of the experiment(From [11])

As mentioned above there is another type of experiment for testing grav-ity at short distances. Here so-called high frequency techniques are used.The torsion balance experiment discussed, has limits for the minimum prac-tical test mass separation due to background effects. They become moreproblematic with decreasing distance. However, experiments using highfrequency techniques do alow us to operate at smaller test mass separa-tion. A research group at Colorado used this technique. We shall discusstheir experiment briefly.

The active components of the apparatus used by the Colorado group is pic-tured in figure 8. The experiment uses a tungsten torsional oscillator (thedetector in the picture). The source of gravitational field is a tungsten vibrat-ing cantilever (the source mass in the picture). The source and detector areseparated by a stiff conducting shield to eliminate background forces due toelectrostatics and residual gas in the vacuum chamber surrounding the ex-


periment. A planar test mass geometry is chosen to concentrate as muchmass as possible at the scale of interest. This is done because of the shortrange of the Yukawa interaction. When you would use a thicker device, theNewtonian interactions would be relatively stronger than the Yukawa forceusing the thin plate, because this force is dominated by nearby mass dueto its exponential character. The best test masses for Yukawa interactionsare heavy thin plates concentrating as much mass as possible at the clos-est range. Therefore a thin plate of tungsten is used. The choice of thismaterial is due to its relatively high density.

Similar to the Eot-Wash group, the Colorado group used a null geometryexperiment with respect to 1

r2 forces. That is, one that should result in azero output for Newtonian gravity if all is well (but we hope it is not!). TheNewtonian gravity exerted by the detector on the reed almost cancels theforce exerted by the reed on the detector. However, this is not the casefor the Yukawa interactions, which become stronger with increasing dis-tance. When the vibrating reed comes closer to the detector there mightbe a small Yukawa force exerted that would influence the oscillating detec-tor. We might be able to measure this distortion. The extensive apparatusarrangement is pictured schematically in figure 9.The attractor formed by the tungsten reed of size 35 mm × 7 mm × 0.305mm was brought into vibration at the natural resonance frequency of thedetector mass, which was around 1 kHz. The detector consists of twocoplanar rectangles joined along their central axes by a short segment.The attractor was positioned so that its front end was aligned with the backedge of the detector rectangle and a long edge of the attractor was alignedabove the detector torsion axis. The attractor, detector and electrostaticshield were mounted on separate vibration-isolation stacks to minimize anymechanical couplings, as can be seen from 9. While our board vibrates inthe resonant mode of interest, it induces a torque on the detector oscillator,causing the detector to counter-rotate in a torsional mode around the axisthat can be seen in . They searched for distortions of the vibration modesafter corrections for unintended effects such as thermal noise. When anextra signal in the oscillation is noticed this might be due to Yukawa inter-action. The minimal test mass separation was 108 µm.


Figure 9: Apparatus (From page 21 of [10])

7.4 Results

7.4.1 Eot-Wash group

The Eot-Wash group found no deviation from Newtonian gravity and inter-pret these results as constraints on extensions of the Standard Model thatpredict Yukawa forces. They set a constraint on the largest single extra di-mension (assuming toroidal compactification and that one extra dimensionis significantly larger than all the others) of R ≤ 160 µm, and on two equal-sized large extra dimensions of R ≤ 130 µm. Yukawa interactions with α ≥1 are ruled out with a confidence level of 95% for λ ≥ 197 µm. We can seethis graphically in figure 10.

7.4.2 Colorado group

The null results from the experiment taken at the minimal separation wereturned into constraints on α and λ using a maximum-likelihood estimation.For various assumed values of λ, the expected Yukawa force was cal-


culated numerically 400 times, each calculation using different values forexperimental parameters that were allowed to vary within their measuredranges. A likelihood function was constructed from these calculations andwas used to extract limits on α from the results with a confidence levelof 95%. This leads to a constraint for the largest extra dimension (againassuming toroidal compactification and that one extra dimension is signif-icantly larger than all the others) of R ≤ 108 µm,The constraints from theColorado group on α and λ are graphically represented in figure 10.

Figure 10: The results for the Yukawa constraints from the Eot-Wash groupare graphed in red. Figure from page 29 of [5]

8 ASTROPHYSICAL BOUNDS 39

8 Astrophysical bounds

In this section we consider bounds placed on the size of the extra dimen-sions by looking at stellar objects. In addition we hope to say somethingabout solving the hierarchy problem. For instance, is M4+n ∼ 1TeV notexcluded? The chosen objects are the Sun, red giants and a supernovanamed SN1987A.

The reason why we look at these objects is that they have very high tem-peratures and we expect KK gravitons to be produced, which can escapeto higher dimensions by several reactions, to be discussed later on. Thosegravitons have momentum and therefore carry energy, which they take withthem while escaping into the extra dimensions. After escaping, they have avery small probability of returning to the brane , since the brane only coversa small region of the higher dimensions. For more information on branesreview section 6.2.

8.1 Cross section of graviton

When a reaction between SM-particles takes place, there is a probabilitythat a graviton is emitted, which can escape into the extra dimensions (i.e. aKK mode) [12]. In particle physics we usually do not talk about probabilities,instead we discuss cross sections. Let’s start with calculating the crosssection of one graviton, σ1. We want to get some sort of probability ofa graviton being emitted. Each KK mode is very weakly coupled ∼ 1

M4.

In a diagram this looks like figure 11. Therefore the probability at whichgravitons are being emitted by a certain reaction involving SM-particles, isgiven by

Figure 11: Coupling of a graviton to a Standard Model particle.


σ1 ∼∣∣∣∣ 1M4

∣∣∣∣2 =(

1M4

)2

(83)

To get the total cross section of the gravitons, we consider the number ofKK modes possibly emitted, when a reaction between SM-particles takesplace, with energy E. The KK modes have momentum p in the extra dimen-sions∼ b

R (see section 3.2), and so bR . E. First consider an 1-dimensional

sphere with radius E. Since p is quantized with spacings of 1R , the number

of KK modes would be ER. This means, for an n-dimensional sphere withradius E, that the number of KK modes is given by

#KKmodes ∼ (ER)n (84)

Combining (84) with σ1 gives

σ ∼ (ER)n

(1

M4

)2

(85)

If we look at possible graviton production by a stellar object, first of all welook at the total energy emitted. This is usually associated with ε, which isthe energy emitted per unit time per unit mass. Secondly we look at εgrav,the energy lost to gravitons per unit time per unit mass. This εgrav dependslinearly on σ [12]. The next thing, to take into account, is the temperatureT of the object. Temperature is associated with energy by setting kB = 1,where kB is Boltzmann’s constant. We can even write T ∼ E. This can beexplained by looking at the Planck spectrum, which tells us about the par-ticles a black body radiates. It gives us a relation between the frequencyν and the intensity of electromagnetic radiation. This depends on the tem-perature of the black body. The Planck spectrum looks like figure 12. Thefrequency corresponding to the maximum is given by Wien’s displacementlaw [16]

ν =2.821kBT

h(86)

We can rewrite this to

E = hν = 2.821kBT (87)


And by setting kB = ~ = 1 and neglecting the numerical factor we obtain

E ∼ T (88)

If T � R−1, KK modes cannot be produced. If T � R−1, they can be pro-duced. The latter case gives a number of KK modes that can be produced[12]. Now by use of T ∼ E and (79), we can rewrite (85) and obtain

σ ∼ Tn

Mn+24+n

(89)

This is exactly what we need, to say something about εgrav, since it dependson σ. The exact derivation of the cross sections of the relevant processestaking place in stellar objects, which can produce gravitons, is beyond thescope of this paper. It can be found in [12]. We just state the relevant crosssections:

• Gravi-Compton scattering: γ + e → e + grav

σv ∼ δe2 Tn

Mn+2(4+n)

β2 (90)

Figure 12: Planck spectrum of a black body for different values of T [16]


• Gravi-brehmstrahlung: (e + Z → e + Z + grav)

σv ∼ δZ2e2 Tn

Mn+2(4+n)

(91)

• Graviton production in photon fusion: γ + γ → grav

σv ∼ δTn

Mn+2(4+n)

(92)

• Gravi-Primakoff process: γ+EM field of nucleus Z → grav

σv ∼ δZ2 Tn

Mn+2(4+n)

(93)

• Nucleon-Nucleon Brehmstrahlung: N +N → N +N + grav N can bea neutron or a proton.

σv ∼ (30millibarn)× (T

M(4+n))n+2 (94)

Where δ = 116π and β = v

c . We have finally obtained enough information totake a closer look to three different stellar objects: the Sun, red giants andSN1987A.

8.2 The Sun and red giants

The first stellar object that allows us to place a bound on M4+n is the Sun.The Sun has temperature ∼ 1keV and

ε ∼ 10−45TeV (95)

The relevant particles are protons, electrons and photons [12]. To considerεgrav we let ε & εgrav. In words this means that the energy loss by gravitonscan never exceed the normal energy loss. This looks like a nonsense de-mand, but in this way we can place a lower bound on M4+n and it can beonly greater than this bound.


The only two relevant processes, by which gravitons are being producedin the Sun, are Gravi-Compton scattering and photon fusion. There are nohigh-Z nuclei, so Gravi-brehmstrahlung is excluded. The Gravi-Primakoff iscomparable to photon fusion, but in this case the latter is more dominantthan the former. And last, Nucleon-Nucleon Brehmstrahlung is excluded,since the temperature of the Sun is too low [12]. Now, let’s consider therelevant processes.

• Gravi-Compton scattering: γ + e → e + grav gives us

εgrav ∼ 4παδTn+5

mpmeMn+24+n

(96)

By use of (95) and T ∼ 1keV , we can rewrite this to

M4+n & 1010−9nn+2 TeV (97)

n 2 3 4 5 6M4+n(TeV ) 10−2 10−4 10−5 10−5 10−6

R(m) 10 10−3 10−5 10−6 10−7

The highest lower bound is given by n = 2 and reads M6 & 10−2TeV . Thisbound does not exclude M4+n ∼ 1TeV . It does give a possible radius tothe extra two dimensions of R ∼ 10m. This is not a problem, since this isthe upper bound for R and R can only be smaller, as it has to be.

• Graviton production in photon fusion: γ + γ → grav gives us

εgrav ∼ δTn+7

ρMn+24+n

(98)

Which gives a lower bound of

M4+n & 1012−9nn+2 TeV (99)

n 2 3 4 5 6M4+n(TeV ) 3 ∗ 10−2 10−3 10−4 10−5 10−6

R(m) 1 10−4 10−5 10−6 10−7


For n = 2, Mn+4 can still be low enough, but notice that this lower bound isslightly higher and so is a stronger bound.

Now we turn to red giants. The only difference in this case is the tempera-ture T ∼ 10keV [12]. This is only slightly higher than the sun’s temperatureand so the same two processes as with the sun apply to red giants. Wewill not repeat all the calculations and we will just conclude that for all nM4+n ∼ 1TeV is still a good possibility.

8.3 SN1987A

As we have seen for the Sun and red giants, M4+n ∼ 1TeV has not yetbeen in danger. To push up the lower bound on M4+n we need to look atstellar objects with higher temperature. Therefore, the collapse of the ironcore of SN1987A is useful for our search to put a strong bound on M4+n.During the collapse the energy emitted was

ε ∼ 10−44TeV (100)

The temperature T of the object is being estimated at 30 − 70MeV [13].To produce a lower bound, we will choose T ∼ 30MeV . The two mainprocesses which took place are, Nucleon-Nucleon Brehmstrahlung and theGravi-Primakoff process [12]. We will only consider the former, since thiswill provide us the strongest bound

M4+n & 1015−4.5n

n+2 TeV (101)

With this we can calculate the lower bound on M4+n and upper bound onR.

n 2 3 4 5 6M4+n(TeV ) 30 2 3 ∗ 10−1 8 ∗ 10−2 3 ∗ 10−2

R(m) 10−6 10−9 6 ∗ 10−11 8 ∗ 10−12 2 ∗ 10−12

The bound for n = 2 is somewhat above 1TeV . The interpretation of thisbound is a bit more tough. The authors of [12] argue that M6 ∼ 30TeVis still consistent with a string scale ∼ fewTeV . While the authors of [13],who use a different technique and find M6 ∼ 50TeV , argue this is too largeto be measured now or in the nearby future.

From this astrophysical perspective we can draw some conclusions aboutthe viability of solving the hierarchy problem and the actual measurement


Figure 13: Upper bounds placed on the size of the possible extra dimen-sions by astrophysics. The red and green line represent Gravi-Comptonscattering, respectively photon fusion, in the Sun. The blue line comesfrom Nucleon-Nucleon Brehmstrahlung in SN1987A.

of any extra dimensions in the laboratory. The results are combined andcan be seen in figure 13. The strongest bound on M4+N is given to us byobservations on SN1987A. For n ≥ 3, M4+n ∼ 1Tev is not excluded. Forn = 2 the analysis becomes a bit more tricky. It is a matter of perspectiveand assumptions if n = 2 can be excluded. So for now, we will not excludethis possibility either. This means the LHC might provide answers in solvingthe hierarchy problem. Nevertheless, finding deviations on the gravitationalforce, by sub-mm tests, does seem implausible. The upper bound on thesize of the extra dimensions reads R . 10−6m, this is far beyond the reachof tests described in section 7.

9 CONCLUSION 46

9 Conclusion

From the short-range gravity experiments we found no evidence for viola-tions of Newtonian gravity due to Yukawa interaction. From this we can onlyexclude some possibilities by setting upper bounds for the size of the extratoroidal compactified dimensions. From the torsion balance experimentswe found the strongest bound to be R ≤ 160 µm. The High frequencyexperiment showed an even stronger bound of R ≤ 108 µm. From the as-trophysical considerations we found the strongest bound to be R . 10−6m,for n = 2. For higher n, R will only decrease further.

For the future there is still a lot of work to do. The torsion-balance experi-ment group try to further improve their apparatus. However, this will be withof lot of modifications. Among other things they will use a larger numberof smaller sized holes. They hope to provide good results for length scalesdown to 50 µm. The high frequency group hope to probe this same rangeof interest. An experiment is planned with a stretched membrane shield inplace of a sapphire plate. If the backgrounds can be controlled, this exper-iment could improve limits for R to fall between 10 - 50 µm.

From astrophysics we saw that observations on stellar objects, show noinconsistency with M4+n ∼ 1TeV , the manner in which ADD try to solvethe hierarchy problem. Even the strongest bound, obtained by observa-tions on SN1987A for n = 2, on M4+n can still be considered within therange of ∼ 1TeV . This bound reads M6 & 50TeV . We have to take intoaccount that the bounds from astrophysics are insecure. How insecureexactly is hard to tell, but it is comforting that M4+n ∼ fewTeV is not ex-cluded yet. This means future particle colliders could start measure signsof extra dimensions. Nevertheless, It seems improbable for sub-mm tests,to find deviations on the gravitational force. It does look good though thatboth ways (i.e. sub-mm tests and astrophysics) of considering deviationshave not contradicted each other. Sub-mm tests have not shown deviationsyet and from astrophysics we know, that deviations cannot be seen at thedistances these tests measure.

REFERENCES 47

References

[1] Lisa Randall. Warped Passages: Unraveling the mysteries of the uni-verse’s hidden dimensions. Page 11 - 21. New York 2005

[2] The Elegant Universe, Brian Greene. Vintage Books, 2000

[3] N. Arkani-Hamed, S. Dimopoulos and G. Dvali, ”The Hierarchy Prob-lem and New Dimensions at a Millimeter.” Phys.Lett. B429 (1998) 263-272

[4] A. Kehagias and K. Stefsos, ”Deviations from the 1/r2 Newton law dueto extra dimensions.” Phys.Lett. B472 (2000) 39-44 hep-ph/9905417

[5] C.D. Hoyle et al. Sub-millimeter Tests of the Gravitational Inverse-square Law. Phys.Rev. D70+ (2004) 042004 hep-ph/0405262

[6] Joshua C. Long et al. New Experimental Limits on Macroscopic ForcesBelow 100 Microns. Nature 421, 922-925 (2003) hep-ph/0210004

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[9] http://www.npl.washington.edu/eotwash/shortr.html

[10] Joshua C. Long, Hilton W. Chan, John C. Price. Experimental Sta-tus of Gravitational-Strength Forces in the Sub-Centimeter Regime.Nucl.Phys. B539 (1999) 23-34 hep-ph/9805217

[11] http://spot.colorado.edu/ pricej/gravity.html

[12] N. Arkani-Hamed, S. Dimopoulos and G. Dvali, ”Phenomenology, As-trophysics and Cosmology of Thearies with Sub-Millimeter Dimen-sions and TeV Scale Quantum Gravity.” Phys.Rev. D59 (1999) 086004hep-ph/9807344

[13] S. Cullen and M. Perelstein, ”SN1987A Constraints on Large CompactDimensions.” Phys.Rev.Lett. 83 (1999) 268-271 hep-ph/9903422

REFERENCES 48

[14] I.S. gradshteyh, I.M. Ryzhik. ”Table of Integrals, Series and Products”San Diego 2000

[15] Borwein, J. M. and Borwein, P. B. ”Poisson Summation.” Page 36-40 inPi and the AGM: A Study in Analytic Number Thery and ComputationalComplexity. New York: Wiley, 1987.

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